While the singular cohomology of a space has always a product structure, this is not true for the singular homology of a space. Nevertheless, it is possible to construct such a structure for an oriented manifold of dimension. This is the so-called intersection product. Intuitively, one can describe it as follows: given classes and, take their product and make it transversal to the diagonal. The intersection is then a class in, the intersection product of and. One way to make this construction rigorous is to use stratifolds. Another case, where the homology of a space has a product, is the loop space of a space. Here the space itself has a product by going first the first loop and then the second. There is no analogous product structure for the free loop space of all maps from to since the two loops need not have a common point. A substitute for the map is the map where is the subspace of, where the value of the two loops coincides at 0 and is defined again by composing the loops.
The Chas–Sullivan product
The idea of the Chas–Sullivan product is to now combine the product structures above. Consider two classes and. Their product lies in. We need a map One way to construct this is to use stratifolds to do transversal intersection. Another approach starts with the collapse map from to the Thom space of the normal bundle of. Composing the induced map in homology with the Thom isomorphism, we get the map we want. Now we can compose with the induced map of to get a class in, the Chas–Sullivan product of and .
Remarks
As in the case of the intersection product, there are different sign conventions concerning the Chas–Sullivan product. In some convention, it is graded commutative, in some it is not.
Furthermore, we can replace by. By an easy variation of the above construction, we get that is a module over if is a manifold of dimensions.
The Serre spectral sequence is compatible with the above algebraic structures for both the fiber bundle with fiber and the fiber bundle for a fiber bundle, which is important for computations.
The Batalin–Vilkovisky structure
There is an action by rotation, which induces a map Plugging in the fundamental class, gives an operator of degree 1. One can show that this operator interacts nicely with the Chas–Sullivan product in the sense that they form together the structure of a Batalin–Vilkovisky algebra on. This operator tends to be difficult to compute in general. The defining identities of a Batalin-Vilkovisky algebra were checked in the original paper "by pictures." A less direct, but arguably more conceptual way to do that could be by using an action of a cactus operad on the free loop space. The cactus operad is weakly equivalent to the framed little disks operad and its action on a topological space implies a Batalin-Vilkovisky structure on homology.
Field theories
There are several attempts to construct field theories via string topology. The basic idea is to fix an oriented manifold and associate to every surface with incoming and outgoing boundary components an operation which fulfills the usual axioms for a topological field theory. The Chas–Sullivan product is associated to the pair of pants. It can be shown that these operations are 0 if the genus of the surface is greater than 0 A more structured approach gives the structure of a degree open-closed homological conformal field theory with positive boundary. Ignoring the open-closed part, this amounts to the following structure: let be a surface with boundary, where the boundary circles are labeled as incoming or outcoming. If there are incoming and outgoing and, we get operations parametrized by a certain twisted homology of the mapping class group of.
